Step by step
The question
Question: find f(−3) for f(x) = x² if x < 0, 2x + 1 if x ≥ 0.
f(x) = x2 if x < 0
= 2x + 1 if x ≥ 0
This is a piecewise function.
It has 2 rules.
Which rule we use depends on x.
Each x uses only one rule.
piecewise: a different rule for different x's
each x uses only one rule
Careful!
Not the second rule!
It is only for x greater than or equal to 0, and −3 is not.
Careful: not f(−3) = 2(−3) + 1 = −5 ✗
1 Find the right piece
Let's see where each rule works on a number line.
So we use this rule: x².
2 Put −3 in
Put −3 in place of x in x².
We write it in brackets.
f(−3) = (−3)2
(−3)² means −3 times −3.
Minus times minus is plus.
(−3)2 = (−3) × (−3) = 9
That is 9.
f(−3) = 9
So f(−3) = 9.
3 Check
Let's check: does −3 fit each rule's x's?
Only one must say yes.
first rule: −3 < 0 yes
second rule: −3 ≥ 0 no ✗
Only the first rule fits.
So f(−3) is right!
Let's look back.
First: find the right piece.
Then: put −3 in.
Last: check.
Answer
f(−3) = 9